As an object orbits another object, periapsis is that point at which the orbiting object is closest to the object being orbited; apoapsis is that point at which the orbiting object is farthest from the object beinn orbited.

Planetary Orbits
In the elliptical orbit, the orbited object will sit at one focus; with nothing present at the other focus. As a planet approaches periapsis, the planet will increase in velocity. As a planet approaches apoapsis, the planet will decrease in velocity.

As the object moves, it falls toward the orbited object. However it moves so quickly that the curvature of the orbited object will fall away beneath it.

A force, gravity, pulls the object into a curve as it attempts to fly off in a straight line.

As the object falls, it moves sideways fast enough to miss the orbited object. This understanding is particularly useful for mathematical analysis, because the object's motion can be described as the sum of three one-dimensional orbits around a gravitational center.

Newton's Laws of Motion
For a system of only two bodies that are only influenced by their mutual gravity, their orbits can be exactly calculated by Newton's laws of motion and gravity. Briefly, the sum of the forces will equal the mass times its acceleration. Gravity is proportional to mass, and falls off proportionally to the square of distance.

To calculate, it is convenient to describe the motion in a coordinate system that is centered on the heavier body, and we can say that the lighter body is in orbit around the heavier body.

An unmoving body that's far from a large object has more energy than one that's close. This is because it can fall farther. This is called "potential energy" because it is not yet actual.

With two bodies, an orbit is a flat curve. The orbit can be open (so the object never returns) or closed (returning), depending on the total kinetic+potentialenergy of the system.

An open orbit has the shape of a hyperbola (or in the limiting case, a parabola); the bodies approach each other for a while, curve around each other around the time of their closest approach, and then separate again forever. This is often the case with comets that occasionally approach the Sun.

A closed orbit has the shape of an ellipse (or in the limiting case, a circle). The point where the orbiting body is closest to Earth is the perigee, called periapsis (less properly, "perifocus" or "pericentron") when the orbit is around a body other than Earth. The point where the satellite is farthest from Earth is called apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides, sometimes called the major-axis of the ellipse. It's simply a line drawn through the longest part of the ellipse.

Orbiting bodies in closed orbits repeat their path after a constant period of time. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be
formulated as follows:

The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of the ellipse. Therefore the orbit lies in a plane, called the orbital plane. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits around particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or, synonymously, periselene and aposelene). An orbit around any star, not just the Sun, has a periastron and an apastron

As the planet moves around its orbit during a fixed amount of time, the line from Sun to planet sweeps a constant area of the orbital plane, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time."

For each planet, the ratio of the 3rd power of its average distance to the Sun, to the 2nd power of its period, is the same constant value for all planets.

Except for special cases like Lagrangian points, no method is known to solve the equations of motion for a system with three or more bodies.

Instead, orbits can be approximated with arbitrarily high accuracy. These approximations take two forms.

One form takes the pure elliptic motion as a basis, and adds perturbation[?] terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moon, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation.

The "differential equation" form is sometimes used for scientific or mission-planning purposes. It calculates the position of the objects a tiny time in the future, then repeats. According to Newston's laws, the sum of all the forces will equal the mass times its acceleration (F=MA). The perturbation terms are much easier to describe in this form. However tiny arithmetic errors from the limited accuracy of a computer's math accumulate, limiting the accuracy of this approach.

Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large objects have been simulated.

A body moving in a 3-dimensional space has 6 degrees of freedom (3 for its position in the 3-dimensional space, and 3 for its velocity in that space). Its orbit is exactly determined by 6 independent parameters. Usually the following orbital parameters are used:

Note that the definition of "mean radius" used by some sources can be somewhat different from that listed above; if you average the radius over time for one orbit or over central angle (true anomaly) then the average distance is a function of both semimajor axis and eccentricity.

If some part of a body's orbit enters an atmosphere, its orbit can decay because of drag. Each periapsis the object scrapes the air, losing energy. Each time, the orbit grows more eccentric (less circular) because the object loses sideways motion. Eventually, the periapsis of the orbit drops low enough that the body hits the surface or burns in the atmosphere.

The bounds of an atmosphere vary wildly. During solar maximums, the Earth's atmosphere causes drag up to a hundred kilometers higher than during solar minimums.

Some satellites with long conductive tethers can also decay because of electromagnetic drag from the Earth's magnetic field. Basically, the wire cuts the magnetic field, and acts as a generator. The wire moves electrons from the near vacuum on one end to the near-vacuum on the other end. The orbital energy is converted to heat in the wire.

Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails. These forms of propulsion require no propellant or energy input, and so can be used indefinitely. See statite for one such proposed use.

In the case of gravity, scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence the duration of one revolution remains the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the earth.